Historically, the first Price of Anarchy (PoA) results for congestion games focused on pure equilibrium (e.g. [1]}, [2]}, [3]}), and were then extended to mixed Nash equilibrum (e.g. [4]}, [5]}, [6]}).
The barriers to equilibrium computation in congestion games inspired the development of techniques, in particular Roughgarden's smoothness framework, for bounding the PoA with respect to tractable solution concepts such as correlated equilibrium (e.g. [7]}, [8]}). Our results show that mixed Nash equilibrium in congestion game is intractable, further motivating the latter line of work on analyzing tractable solution concepts.
